In addition, assuming that the Markov chain \((X_n)_{n\ge 0}\)  is an adapted stochastic process defined on a filtered probability
space \((\Omega ,\lbrace \mathcal {F}_m\rbrace _{m\ge 0},\lbrace \mathbb {P}_x\rbrace _{x\in E\cup \lbrace \partial \rbrace })\) , it follows from [1]}
and [2]} that, under (REF ), for all \(x\in E^{\prime }\)  where \(E^{\prime }:=\lbrace y\in E:\eta (x)>0\rbrace \) , the limit
\(\mathbb {Q}_x(A)=\lim _{n\rightarrow +\infty }\mathbb {P}_x(A\mid n<\tau _\partial )\) 
